Chapter III: Computational Electrohydrodynamic Lithography of Dielectric Films 44
3.5 Nonlinear Optimization and Results
Algorithm 1Broyden–Fletcher–Goldfarb–Shanno algorithm
1: procedure BFGS
2: Compute and storage ∇DJ|(0) . initialize BFGS
3: Line search to find α= argminJ(D(0)+αG)
4: StorageS(1)=αG
5: UpdateD(1) =D(0)+S(0)
6: Initialize B−1(0)=I
7: Initialize k= 1
8: while objectiveJ(D(k))>tolerancedo . BFGS iteration loop
9: Compute and storage ∇DJ|(k) 10: ComputeY =∇DJ|(k)− ∇DJ|(k−1)
11: Rank-2 symmetric update on the inverse of approximated HessianB−1
B−1(k)=B−1(k−1)+S>(k−1)MY +Y>M B−1(k−1)Y
(S>(k−1)MY)2 S(k−1)S>(k−1)M
− 1
S>(k−1)MY
hB−1(k−1)Y S>(k−1)M +S(k−1)Y>M B−1(k−1)i
12: ComputeG=B−(k)1[−∇DJ|(k)]>
13: Line search to findα= argminJ(D(k)+αG)
14: StorageS(k)=αG
15: UpdateD(k+1)=D(k)+S(k)
16: Shift D(k+1)
17: k→k+ 1
of thickness H0, the equality
1>MHk=1>M(H01) (3.116) holds for all subsequent film states. If total mass of the desired target film profileHover the periodic domain is different from the initially flat state H01, then it is impossible for any film stateHk to converge to targetH no matter how we update the electrode topography D, i.e. target H is unfeasible according to the constraint. In practice, the discrete values of target film profile is generated by a user-specified input Hin, for example, the heart-like protrusion reconstructed from the raster height map shown in figure3.6, is likely not to have an identical total mass withH01. Preprocessing on these user-generated inputs is required to ensure feasibility of the target film profileH used in the optimization procedure. There are infinitely many ways to correct Hin. We prefer shifting Hin by a constant value everywhere,
H =Hin+H0− 1>MHinput 1>M1
1, (3.117)
so that the geometric details of user inputHin is not compromised too much.
-0.1 0 0.1 0.2
Step 5 Step 10
Step 15 Step 20 Step 25
Step 30 Step 35
Step 0
Step 40
Figure 3.11: Sequence of electrode topography profiles 1−D on a periodic square domain[0,4.5]×[0,4.5]at iteration step0,5,...,40of the BFGS optimization process;
the nonlinear EHL evolution equation is discretized with 602 elements and 200 evenly spaced time steps till the final timeτ = 4.2.
We also apply the shifting transformation (3.117) to the discrete electrode topography D after the BFGS update in step 15 for a different reason. The purpose of such shift is to maintain a fixed reference height (e.g., average height) on the electrode topography throughout the entire optimization process and to prevent pathological scenario where electrode topography D being optimized may drift infinitely far away from the film.
Validation on a target film shape of uniform thickness
We verify our implement of the optimization algorithm against the follwing test problem:
find the optimal electrode topography D(X) for a target film profile of uniform thick- ness, which happens to be identical to the initial condition H = H01 due to volume conservation. It is one of the few cases for which exact analytic solution of the optimal electrode topography, i.e. a flat electrodeD=1, is known. Recall from (3.21) that the free energyF[H, D]of the EHL system is always non increasing. Any nonflat electrode would necessarily deform an initially flat film and hence cannot produce another flat state of the same volume at a later timeτ >0. If we take the constraint on the spatial average of electrode topography into account, then the optimal pattern must be the unique “do-nothing” flat electrodeD=1.
The convergence to a flat electrode is shown in figure3.11. The test problem is posed
0 20 40 60 80 100 10-9
10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
BFGS iteration step
#|Dmax−1|atτ =τ 4 |Dmin−1| atτ=τ
3J(H, D)
Figure 3.12: The BFGS optimization underlying figure 3.11: the minimum and maxi- mum signed deviation to the analytic solutionD= 1and objective functionalJ(H,D) at final timeτ are plotted for the electrode being optimized at each BFGS step.
on a periodic square domain [0,4.5]×[0,4.5]and for a time interval 0≤τ ≤4.2. The initial film thicknessH0= 0.15is uniform over the entire domain. TheH1-regularization parameter γ = 5×10−4 is used in the discrete objective functional (3.78). We choose the heart-like electrode plotted in figure 3.6 as the initial guess. In the early stage of optimization, features of high spatial frequency such as edges and corners are rapidly damped. We observe that the optimization process picks up large scale spatial variations in the background after the initial damping. This is due to the inherent nonlinearity of the problem being optimized: different spatial modes of electrode patternD(X)are coupled through the nonlinear constraint, i.e. the EHL evolution equation. In figure 3.12 we plot the maximum and minimum signed deviation to the optimal electrode topography D= 1and discrete objectiveJ over 100 BFGS iteration steps on a log scale. Deviation from the analytic solution of the optimal flat electrodeD=1decreases towards zero as the objective J is minimized. At iteration step 40 shown in figure 3.11, the maximum spatial error of the optimal topography is already driven below 0.01. Th overall linear trend suggests an exponential convergence rate which is typical for optimization methods based on gradient descent (Nocedal and Wright,2006).
The discrete cosine transform of 1−D(X) are shown in figure 3.13 as the electrode topography is being optimized at each iteration step. Each block at a grid num- ber (i, j) represents the absolute (real) amplitude of the (i, j)-th spatial harmonics cos(KiXX) cos(KjYY) where KiX and KjY are the i-th and j-th spatial frequencies of in X andY directions, respectively. Unlike a purely diffusive process for which features of higher frequency always receive faster damping, during late stage of the optimiza- tion the(0,2)-,(0,1)-,(2,0)-,(2,2)-modes seem to persist as a group. This indicates that, the diffusive effect introduced from the regularization functional R[D] is not the main driving force behind the optimization. Instead, the approximate Hessian matrix
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Figure 3.13: Discrete Fourier (cosine) transform of the electrode topography1−D(X) at iteration steps shown in figure 3.11. The color and height of a block at grid number (i,j) correspond to the absolute (real) amplitude Aij of the (i, j)-th cosine harmonics cos(KiX) cos(KjY) where Kk is the k-th spatial frequency of interval [0,4.5]. For visual purpose, log10(1 +|Aij|) is plotted instead.
of the objective (excluding the regularization) may chose to amplify a selection of its eigenvectors which result in the persisting pattern in the Fourier space we see in figure 3.13.
To gain more insights and intuitions into the control algorithm, we plot the discrete solutions to the film state H(X, τ), adjoint variableΛ(X, τ) and the constraint force C(X, τ)at the zeroth step of the nonlinear optimization in figure3.14side by side. Film states H(X, τ) in the left-most column are identical to the ones shown in figure 3.7 subject to the top electrode with a heart-like protrusion (see figure 3.6). The adjoint variable Λk in the middle column propagates the discrepancy between the final film profile H(X, τ) and target profile H as the final time condition backwards in time.
The dynamics of the adjoint variableΛ(X, τ) mimics a reverse-diffusion process where sharp features become blurry while being transported backwards in time. Recall from the control equations (3.67) and (3.89) that the objective gradient ∇DJ at the zeroth iteration step is given by the time integration of all the constraint forces Ck shown in the right-most column at each discrete time step. We observe that the constraint forces closer to the final timeτ are orders of magnitude larger than the ones from the early stage and hence dominate the time integration of all constraint forces. In other words, the behavior of film states in the later stage has significantly more impact on the discrepancy
τ = 0
τ = 15τ
τ = 25τ
τ = 35τ
τ = 45τ
τ = 55τ
0 0.05 0.10 0.15 0.20 0.25 0.30 -0.1 0 0.1 0.2 0.3 -0.5-0.4-0.3-0.2-0.1 0×10-2
H(X, τ) orHk Λ(X, τ) or Λk C(X, τ) or Ck
Figure 3.14: Solutions to the film state H(X, τ) (left column), the adjoint Λ(X, τ) (middle column) and the negative constraint force−C(X, τ) (right column) on a peri- odic computational domain[0,4.5]×[0,4.5]computed from the discrete state, adjoint and control equations (3.75), (3.94) and (3.89). Target film profile H(X) = H0 is identical to the initial flat state. Snapshots with elevation contour (black line) are taken at time stampsτ = 0, τ /5, ..., τ (from top to bottom) where final time τ = 4.2.
0 0.05 0.10 0.15 0.20
0 0.05 0.10 0.15 0.20 -0.6 -0.4 -0.2 0 0.2
(b)
(a) (c)
Figure 3.15: (a) Target film profileH(X)compared to (b) the final film stateH(X, τ) obtained under (c) the optimal electrode topography 1−D(X) found by nonlinear optimization on a square domain [0,4.5]×[0,4.5].
between the final film profileH(X, τ)and target shapeH. This is expected because the evolution equation of the EHL system is dissipative under a time-independent electrode with a static geometry and a constant applied voltage. The memory of previous film states beyond certain time period is eventually lost due to dissipation in the system.
Unless time-dependency is restored in the electrode (voltage distribution, topography, etc.) which would result in extra terms in the control equations, the best strategy to guide the evolving film profileH(X, τ) into the target shapeH(X), suggested by our control algorithm, is to promote the desired convergence right before the final time τ rather than to achieve it earlier.
Optimal electrode design for achieving a heart-like film pattern
We apply the control algorithm to obtain the optimal electrode design for a specific film pattern, i.e. the uniform elevation of a filled heart-like shape at the center of a square.
The simulation and nonlinear optimization are performed on a square domain of edge length 4.5, discretized by 502 Q9 Lagrange finite elements. The heart-like target film profile H(X) is shown in figure 3.15(a). The goal is to look for an optimal electrode topography functionD(X) under which the evolving film state H(X, τ) converges to the target H(X) at final time τ = 4.2. We start the search of the optimal design with an initial guess of a uniform electrode D =1. The H1-regularization parameter γ = 7.5×10−4is used to preserve certain level of smoothness inDthroughout the entire process. After about 100 BFGS iteration steps, the objectiveJ reduces to about6.5%
of its original value produced by the initial guess and the search therefore terminates.
The resulting optimal electrode pattern and the corresponding film profile at the final time τ are plotted in figure 3.15. Snapshots of intermediate film evolution, backward
adjoint propagation and discrete cosine transform (DCT) of film states are shown in the left, middle and right columns of figure 3.16, respectively.
The optimal design solved by the control algorithm seems to suggest an interesting design principle in favor of separating spatial scales. Small-scale features of the optimal electrode shown in figure 3.15(c), e.g., ridges and horns, as usual immediately trigger localized growth resembling the boundary of the heart pattern. However unlike the case of naive electrode designs such as the one in figure 3.6, these structures do not reinforce themselves into narrower pillars. Instead, the subsequent growths are arrested by interface deformations induced by the large-scale sinusoidal oscillations in the optimal electrode topography. As shown in the left column of figure3.16, while the central void enclosed by the narrow ridges are being filled, four weak bumps also start to appear at corners of the square domain. Due to conservation of mass, if the dielectric film is thickening at the center and corners, it must drain liquid from the region in between, exactly where the prior development of narrow ridges occurs. The competition between the growth of large and small features is made manifest by the discrete cosine transform ofH(X, τ)in the right column of figure3.16. The rapid development of high frequency modes (e.g., blocks in white) is suddenly choked by the emergence of low frequency oscillations (e.g., blocks in yellow, orange and red) during the time interval between 4τ /5 and the final time τ. Hence a smart combination of sharp and blunt features in the optimal electrode topography computed by our optimization algorithm not only trigger pattern formations in dielectric film at multiple resolutions but also regulate the growth rate of of spatial modulations at both large and small scales as well as the temporal order at which these modulations emerge in order to achieve a complex pattern such as the heart-like pillar.
Lastly, we briefly discuss the role of regularization, particularly the H1-regularization (3.50). In our specific example of optimizing for a heart-like film pattern, the choice of regularization parameter γ seems to have minor effects on the nonlinear search.
Although from the final film statesH(X, τ)plotted in the the first row of figure3.17, it is slightly more difficult to reproduce corners and edges of the heart-like pattern with a higher regularization value. This is expected because a strongerH1-regularization would penalize sharp electrode features: the initial growth of film patterns triggered by these sharp features can only be corrected later by interface modulations emerging on much larger scales due to disspative nature of the EHL system. In future work, it is interesting to explore other choices of objective functional and regularization, for example, the use of l1-norm instead l2, i.e. J =RΩ|H−H|dΩ, or distributed ones, i.e. J and R are defined independently on disjoint subdomains (Barker, Rees, and Stoll,2016).
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0 0.05 0.10 0.15 0.20 -0.01 0 0.01 0.02 0.03 0 0.1 0.2 0.3 0.4
H(X, τ) or Hk Λ(X, τ) orΛk DCT ofH(X, τ)−H0
Figure 3.16: Film state H(X, τ) (left column), the adjoint Λ(X, τ) (middle column) and the discrete cosine transform (DCT) of H(X, τ) (right column) computed for a heart-like target profile on a periodic square domain [0,4.5]2. Snapshots are taken at time stampsτ = 0, τ /5, ... , τ (from top to bottom) where final timeτ = 4.2. Legend setup of DCT is identical to the one in figure 3.13.
0.100 0.125 0.150 0.175 0.200 0.225
0.10 0.15 0.20 0.25 0.30
γ = 0.001 γ= 0.002 γ = 0.003
Figure 3.17: Final film state H(X, τ) (top row) at τ = 4.2 and optimal electrode topography1−D(X)(bottom row) on a periodic square domain[0,4.5]2 produced by nonlinear optimizations under differentH1-regularization parameters γ.